Neumann boundary controllability of the Gear–Grimshaw system with critical size restrictions on the spatial domain

Abstract

In this paper we study the boundary controllability of the Gear-Grimshaw system posed on a finite domain (0, L), with Neumann boundary conditions:We first prove that the corresponding linearized system around the origin is exactly controllable in (L 2 (0, L)) 2 when h2(t) = g2(t) = 0. In this case, the exact controllability property is derived for any L > 0 with control functions h0, g0 ∈ H − 1 3 (0, T ) and h1, g1 ∈ L 2 (0, T ). If we change the position of the controls and consider h0(t) = h2(t) = 0 (resp. g0(t) = g2(t) = 0) we obtain the result with control functions g0, g2 ∈ H − 1 3 (0, T ) and h1, g1 ∈ L 2 (0, T ) if and only if the length L of the spatial domain (0, L) belongs to a countable set. In all cases the regularity of the controls are sharp in time. If only one control act in the boundary condition, h0(t) = g0(t) = h2(t) = g2(t) = 0 and g1(t) = 0 (resp. h1(t) = 0), the linearized system is proved to be exactly controllable for small values of the length L and large time of control T . Finally, the nonlinear system is shown to be locally exactly controllable via the contraction mapping principle, if the associated linearized systems are exactly controllable.1991 Mathematics Subject Classification. Primary 35Q53; Secondary 37K10, 93B05, 93D15.

Boundary controllability of a nonlinear coupled system of two Korteweg–de Vries equations with critical size restrictions on the spatial domain

“…As conjectured by Capistrano-Filho et al in [2], indeed we can prove that system (1.1)-(1.7) is controllable if and only if the length L of the spatial domain (0, L) does not belong to a new countable set, i. e., (1.8) L ∈ / F r := π (1 − a 2 b)α(k, l, m, n, s) 3r : k, l, m, n, s ∈ N ,…”

“…Later on, in [4], two boundary controls were considered to prove that the same system is exactly controllable for small values of the length L and large time of control T . Here, we use the ideas contained in [2] to prove that, with another configuration of four controls, it is possible to prove the existence of the so-called critical length phenomenon for the nonlinear system, i. e., whether the system is controllable depends on the length of the spatial domain. In addition, when we consider only one control input, the boundary controllability still holds for suitable values of the length L and time of control T .…”

Local null controllability of a model system for strong interaction between internal solitary waves

“…To our knowledge, this problem has not been addressed in the literature and the existing developments do not allow to give an immediate answer to it. Indeed, the controllability problem for system (1.1) was first solved when the control acts through the boundary conditions [8,9,17,35]. The results are obtained combining the analysis of the linearized system and the Banach's fixed-point theorem.…”

“…Later on, in [35], the authors proved the local exact boundary controllability property for the nonlinear system (1.1) posed on a bounded interval. Their result was improved in [9], [17] and in [8], where the same boundary control problem was addressed with a different set of boundary conditions. In all those works, except in [17], the results were proved by applying the duality approach and some ideas introduced by Rosier in [36].…”

Internal null controllability of the generalized Hirota-Satsuma system

“…Regarding dispersive systems, we find papers dealing with the boundary controls of either KdV systems on a bounded domain [8,16,4,5] or KdV equations posed on a network [1,7]. Concerning the internal control of dispersive systems, the closest works are [17] where Ingham theorems are used to prove some observability inequalities for Boussinesq systems and [2] where a Carleman estimates approach is used to get the null controllability of a linear system coupling a KdV equation with a Schrödinger equation.…”